Suppose that an insect population density, in thousands, during year n can be modeled by the recursively defined sequence: $$a _ { 1 } = 1 ; a _ { n } = 2.98 a _ { n - 1 } - 0.17 a _ { n - 1 } ^ { 2 } \quad \text { for } n 1$$ Use technology to graph the sequence for n = 1, 2, 3, . . . , 20 . Describe what happens to the population density
Function.
A) The insect population increases every year.
B) The insect population stabilizes near 11.65 thousand.
C) The insect population stabilizes near 12.58 thousand.
D) The insect population stabilizes near 8.45 thousand.

Question

Suppose that an insect population density, in thousands, during year n can be modeled by the recursively defined sequence: $$a _ { 1 } = 1 ; a _ { n } = 2.98 a _ { n - 1 } - 0.17 a _ { n - 1 } ^ { 2 } \quad \text { for } n > 1$$ Use technology to graph the sequence for n = 1, 2, 3, . . . , 20 . Describe what happens to the population density
Function.
A) The insect population increases every year.
B) The insect population stabilizes near 11.65 thousand.
C) The insect population stabilizes near 12.58 thousand.
D) The insect population stabilizes near 8.45 thousand.

Quizplus · Accepted Answer

The recursive formula given models a population that initially grows but eventually stabilizes due to the negative quadratic term, which represents the limiting factors on population growth (like food scarcity or increased competition). By graphing the sequence for $n = 1, 2, 3, \ldots, 20$, it can be observed that the population density approaches a stable value. The correct stable value, near 11.65 thousand, is found by identifying the point at which the population growth rate effectively becomes zero, indicating a stable population size. This is characteristic of logistic growth models, where populations stabilize at a carrying capacity due to limiting factors.

Suppose That an Insect Population Density, in Thousands, During Year a1=1;an=2.98an−1−0.17an−12 for n>1a _ { 1 } = 1 ; a _ { n } = 2.98 a _ { n - 1 } - 0.17 a _ { n - 1 } ^ { 2 } \quad \text { for } n > 1a1​=1;an​=2.98an−1​−0.17an−12​ for n>1

Suppose That an Insect Population Density, in Thousands, During Year $a _ { 1 } = 1 ; a _ { n } = 2.98 a _ { n - 1 } - 0.17 a _ { n - 1 } ^ { 2 } \quad \text { for } n > 1$